# How to solve the singularly perturbed problem below

There is a singularly perturbed problem:

\begin{align} &\epsilon u''(x) + u(x) - u^2(x)=0,\quad x<0<1,\\ & u(0)=1, \quad u(1)=0. \end{align}

where $$\epsilon \ll 1$$ is a tiny parameter. From the numerical result, it has a boundary layer at $$x=1$$ (The graph below is the numerical solution). I would like to find the asymptotic solution with the method of matched asymptotic expansions, but I have no idea how to deal with such a nonlinear problem.

Could anyone give me some hints? Of course, other methods or the exact solution are also available. Here, I post the analytic result that I get until now (haven't finished yet). Consider the matched asymptotic expansion.

Step1: Outer Solution

Assume that the solution can be expanded in powers of $$\epsilon$$, namely, $$u(x,\epsilon)=u_0(x)+\epsilon u_1(x) + \mathcal{O}(\epsilon^2)$$ Substituting this into original problem and consider the $$\mathcal{O}(1)$$ equation, that is $$\epsilon=0$$, we get \begin{align} & u_0^2(x)-u_0(x)=0, \\ & u_0(0)=1. \end{align} We know that $$u_0(x)=1$$ is a or solution. From the graph above, it is reasonable.

Step2: Boundary Layer

Based on the numerical result, there is a boundary layer at $$x=1$$, so the stretching transformation we take is $$\bar x = \frac{x-1}{\delta}.$$

From the stretching transformation and the chain rule, we have that $$\frac{d}{dx}=\frac{1}{\delta}\frac{d}{d \bar x}.$$

Let the $$U(\bar x)$$ be the solution using the stretching transformation, then the original problem becomes \begin{align} &\frac{\epsilon}{\delta^2}U''(\bar x)+U(\bar x)-U^2(\bar x)=0, \\ &U(0)=0. \end{align} The boundary condition comes from the fact that $$u(1)=U(\frac{1-1}{\delta})=U(0)=0.$$ The domainant balances illustrates $$\frac{\epsilon}{\delta^2}=1$$, namely, $$\epsilon = \delta^2.$$ Thus, the transformed problem becomes \begin{align} &U''(\bar x)+U(\bar x)-U^2(\bar x)=0, \\ &U(0)=0. \end{align} After solving the equation above, we may get the solution that includes a parameter $$c$$. Then, matching the solution of boundary layer and outer solution may get the parameter $$c$$, and obtain the final asymptotic solution. But I have no idea how to solve the equation above.

You get a center around $$u=0$$ and a saddle point around $$u=1$$. To get at least one-sided convergence the solution has to follow the level curve of the saddle point, $$ϵu'^2+u^2-\frac23u^2=\frac13.$$ This gives $$u'(1)=\pm\frac1{\sqrt{3ϵ}}$$, giving an IVP for this curve. This solution will not exactly meet $$u(0)=1$$, only come very close to it, with an error of the form $$e^{-c/\sqrtϵ}$$ which is smaller than any power of $$ϵ$$ asymptotically for $$ϵ\to0$$.
• Thanks for your answer. I drew the graph according to your answer, and the solution is $u(x)=sin(\frac{x-1}{\sqrt{3\epsilon}})$. It is indeed a good approximation. This problem, however, has several solutions. One of them is shown in the graph that I added above. So I would like to ask how can I get the asymptotic solution shown in the graph? sorry for the unclear question before. By the way, how can I get the level curve of the saddle point that you post? I study the numerical stuff and have little knowledge about the analytic one. Jun 15, 2021 at 12:38